On the free convection from a horizontal plate

Zusammenfassung Die freie Konvektion an einer beheizten horizontalen Platte wird nach den üblichen Methoden der Grenzschichttheorie untersucht. Weist die wärmeübertragende Plattenseite nach oben, so ist das Problem auf dieser Basis unlösbar, wohl aber findet man eine widerspruchsfreie Lösung, wenn diese Plattenseite nach unten weist. In diesem ist die Nusselt Zahl, die den Wärmeübergang kennzeichnet, proportional der 1/5-ten Potenz der Rayleigh Zahl, welche die Beheizung charakterisiert. Die Übereinstimmung mit dem Experiment ist befriedigend.     

On the viscous dissipation in the boundary layer of a high Prandtl number fluid in laminar flow over a flat plate

Summary An asymptotic expansion for large Prandtl number fluids has been obtained for the recovery factor in a laminar flow over flat plate. The expression underestimates the recovery factor by no more than 1% as long as the Prandtl number is greater than seven.

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Zusammenfassung Die adiabatische Oberflächentemperatur (recovery factor) für ein Fluid mit hoher Prandtl-Zahl wurde für laminare Strömung über eine ebene Platte mit Hilfe einer asymptotischen Entwicklung bestimmt. Der Ausdruck unterschätzt das genaue Resultat um weniger als 1 % für Prandtl-Zahlen höher als 7.

     

On rotational viscous flow through a tube

Zusammenfassung Für die Eigenwerte und Eigenfunktionen eines Eigenwertproblems, auf dasL. Collatz undH. Görtler im Zusammenhang mit einer kürzlichen Untersuchung [3] über Rohrströmung mit schwachem Drall geführt worden sind, werden asymptotische Näherungsformeln angegeben, die insbesondere die höhern Eigenwerte mit grösserer Genauigkeit als in [3] zu berechnen gestatten. Auch zur Ermittlung der niedrigen Eigenwerte wird ein einfaches Verfahren vorgeschlagen. Die Ergebnisse werden numerisch mit denen von [3] verglichen.

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This paper was prepared under a National Bureau of Standards contract with American University and was supported by the Office of Naval Research.     

On the steady viscous flow of a nonhomogeneous asymmetric fluid

We consider a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a C ² boundary. We use a stream-function formulation after the manner of Frolov (Math Notes 53(5–6):650–656, 1993) in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. We show existence to the resulting system using Galerkin method.     

On pulsatile flow of a viscous fluid in a rotating channel

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK ²=ΩL ²/v which is reciprocal of Ekmann Number and frequency parameter σ=αL ²/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK ² the flow near the plates has a multiple boundary layer character.     

On the theory of countercurrent flow in a rotating viscous heat-conducting gas

The countercurrent flow in a gas centrifuge is simulated. Mechanical and thermal methods for its excitation are discussed; thermal restructuring, the thermal control of the velocity field, and a shift in the inversion point are analyzed; and the formation of overtone flows in the rarefaction zone is studied.     

On the flow of a dusty viscous liquid through a circular pipe

Analytical solution of the flow problem of a dusty viscous liquid through a circular pipe in case of axial symmetry is obtained when pressure gradient varies harmonically with time. It is found that the effect of the fine dust is to make the velocity of sedimentation zero and when dust is sufficiently coarse, the effect of the dust is equivalent to an extra frictional force proportional to the fluid velocity.     

Numerical solution of the free‐surface viscous flow on a horizontal rotating elliptical cylinder

The numerical solution of the free‐surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time‐dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free‐surface is determined as part of the solution. For the time‐dependent case a simple time‐marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break‐down occurs. Time‐dependent solutions do not produce the periodic solution after a long time‐scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub‐maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On a perturbation treatment of a model for MHD viscous flow

We discuss the solution of a nonlinear ordinary differential equation that appears in a model for MHD viscous flow caused by a shrinking sheet. We propose an accurate numerical solution and derive simple analytical expressions. Our results suggest that a recent perturbation treatment of the same problem exhibits a pathological behaviour and conjecture its probable cause.     

On the optimal control of viscous incompressible fluid flow

The framework of the Navier-Stokes (N-S) equations is used to study flow past an arbitrary body on whose surface the tangential or normal velocity is under control. The necessary conditions are obtained for the minimum rate of energy dissipation. Exact analytical solutions of the corresponding problems are found for the case of flow past an ellipsoid in the Stokes approximation.     

On initial value problems for the flow in a thin sheet of viscous liquid

Résumé Une couche mince d'un liquide visqueux coule sur un plan horizontal. On établit l'équation différentielle, qui est non-linéaire pour ce mouvement. Pour le cas d'une concentration initiale du liquide à l'origine, nous calculons le développement de la forme de la surface libre. En outre, une solution de similitude est determinée.     

Long cycles passing through a specified edge in a 3-connected graph

We prove the following theorem: For a connected noncomplete graph G, let τ(G): = min{d_G(u) + d_G(v)|d_G(u, v) = 2}. Suppose G is a 3-connected noncomplete graph. Then through each edge of G there passes a cycle of length ≥ min{|V(G)|, τ (G) − 1}. © 1997 John Wiley & Sons, Inc.     

On the control of layered flow of a viscous incompressible fluid within MHD

The control of a viscous incompressible flow within MHD is considered. The phenomenon of internal heat release is explored as applied to a layered plane flow in which the kinetic energy dissipates strictly into heat. The influence exerted by the electromagnetic field of the flowing fluid on the surrounding medium is examined, and methods for flow control in the context of digital deposit field technology are considered.     

On the plane couette flow of a viscous compressible fluid with transpiration cooling

The exact solution for the plane couette flow of a viscous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity profiles and the enthalpy are shown graphically.     

On a finite element method for a compressible viscous flow in a MOCVD reactor

We propose a stable finite element method for approximating the flow of a chemically reacting gas mixture in an MOCVD (metal‐organic chemical vapor deposition) reactor. The flow is governed by the full compressible Navier‐Stokes equations extended by transport equations for the chemical species, the energy equations and the equation of state, together with boundary conditions providing information on the reactor geometry and experimental conditions. The equations form a semilinear system with a constraint for which the corresponding pressure term is not the Lagrangian multiplier. An application of our method to a real world model of growth of GaAs shows the consistency with experimental data. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005